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Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues

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Abstract

For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ2-k (resp. D k-1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are “dual” under ξ2-k to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of D k-1.

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References

  1. Borcherds R. (1998) Automorphic forms with singularities on Grassmannians. Invent. Math. 132: 491–562

    Article  MATH  MathSciNet  Google Scholar 

  2. Bringmann K., Ono K. (2006) The f(q) mock theta function conjecture and partition ranks. Invent. Math. 165: 243–266

    Article  MATH  MathSciNet  Google Scholar 

  3. Bringmann, K., Ono, K.: Dyson’s ranks and Maass forms. Ann. Math. (in press)

  4. Bringmann K., Ono K. (2007) Lifting cusp forms to Maass forms with an application to partitions. Proc. Natl. Acad. Sci. USA 104(10): 3725–3731

    Article  MathSciNet  Google Scholar 

  5. Bringmann, K., Ono, K., Rhoades, R.C.: Eulerian series as modular forms. J. Am. Math. Soc. (in press)

  6. Bruinier, J.H.: Borcherds products on O(2, l) and Chern classes of Heegner divisors. Springer Lecture Notes in Mathematics, vol. 1780. Springer, Heidelberg (2002)

  7. Bruinier J.H., Funke J. (2004) On two geometric theta lifts. Duke Math. J. 125: 45–90

    Article  MATH  MathSciNet  Google Scholar 

  8. Bruinier, J.H., Ono, K.: Heegner divisors, L-functions, and harmonic weak Maass forms. (preprint)

  9. Bump D. (1997) Automorphic forms and representations. Cambridge University Press, Cambridge

    Google Scholar 

  10. Hejhal D.A. (1983) The Selberg trace formula for \({{\rm PSL}(2, \mathbb {R})}\), Lecture Notes in Mathematics, vol. 1001. Springer, Berlin

    Google Scholar 

  11. Iwaniec, H.: Topics in the classical theory of automorphic forms. Grad. Studies in Math., vol. 17. Amer. Math. Soc., Providence, R.I. (1997)

  12. Lewis J., Zagier D. (2001) Period functions for Maass wave forms. I. Ann. Math. 153: 191–258

    Article  MATH  MathSciNet  Google Scholar 

  13. Ono, K.: The web of modularity: arithmetic of the coefficients of modular forms and q-series. Conference Board of the Mathematical Sciences 102. Am. Math. Soc. (2004)

  14. Ono, K.: A mock theta function for the Delta-function. In: Proceedings of the 2007 Integers Conference (to appear)

  15. Petersson H. (1932) Über die Entwicklungskoeffizienten der automorphen Formen. Acta Math. 58: 169–215

    Article  MathSciNet  Google Scholar 

  16. Rademacher H. (1973) Topics in analytic number theory. Springer, New York

    MATH  Google Scholar 

  17. Serre, J.-P.: Formes modulaires et fonctions zêta p-adiques, Springer Lect. Notes in Math., vol. 350. Springer, Berlin, pp. 191–268 (1973)

  18. Serre J.-P. (1976) Divisibilité de certaines fonctions arithmétiques. L’Enseign. Math. 22: 227–260

    MATH  MathSciNet  Google Scholar 

  19. Shimura G. (1994) Introduction to the arithmetic theory of automorphic functions. Princeton University Press, Princeton

    MATH  Google Scholar 

  20. Zagier, D.: Ramanujan’s mock theta functions and their applications [d’après Zwegers and Bringmann-Ono]. Sém. Bourbaki (2007) (in press)

  21. Zwegers, S.P.: Mock theta functions. Ph.D. Thesis, Universiteit Utrecht (2002)

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289, Darmstadt, Germany

    Jan H. Bruinier

  2. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Ken Ono & Robert C. Rhoades

Authors
  1. Jan H. Bruinier
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  2. Ken Ono
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  3. Robert C. Rhoades
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Correspondence to Ken Ono.

Additional information

The second author thanks the generous support of the National Science Foundation, and the Manasse family. The third author is grateful for the support of a National Physical Sciences Consortium Graduate Fellowship, and a National Science Foundation Graduate Fellowship.

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Bruinier, J.H., Ono, K. & Rhoades, R.C. Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues. Math. Ann. 342, 673–693 (2008). https://doi.org/10.1007/s00208-008-0252-1

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  • DOI: https://doi.org/10.1007/s00208-008-0252-1

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